Encyclopedia of Microtonal Music Theory

13mu / tridekamu

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

A term coined in July 2003 by a group of tuning theorists (including Aaron Hunt, Gene Ward Smith, and Joe Monzo), to describe one of a family of terms referring to units of resolution in MIDI tuning, used in electronic music software and computer music software. The prefix specifies the exponent of 2 which describes the number of MIDI tuning units per semitone, and the final "mu" is an acronym for "MIDI unit". In this work the numerical figure is used in preference to the verbal prefix.

At the setting for 13mu pitch-bend resolution, a semitone is divided into 213 = 8192 pitch-bend units. Thus there are 8192 * 12 = 98304 13mus in an "octave", so the 13mu measurement system may be thought of as 98304-edo tuning, with a 13mu being one degree in 98304-edo.

A 13mu is calculated as the 98304th root of 2 -- 98304√2, or 2(1/98304) -- with a ratio of approximately 1:_. It is an irrational number, but is extremely close to the ratio _ : the difference is ~1/_ of a cent, which for all intents and purposes makes the 13mu identical to that ratio. The formula for calculating the 13mu-value of any ratio is:
13mus = log10(ratio) * [ (213 * 12) / log10(2)]
or
13mus = log2r * (213 * 12)
, where r is the ratio.

1.000007051. It is an irrational number, but is extremely close to the ratio 141825:141824 : the difference is only ~ 1/6,000,000

A 13mu is:

exactly 125/12288 (= 0.010172526041666... ~= 1/98) of a millioctave

exactly 25/2048 (= 0.01220703125 ~= 1/82 ) of a cent

exactly 1325/12288 (= 0.107828776041666... ~= 1/9 ) of a türk-sent

exactly 30103/98304 (= 0.306223551432291666... ~= 1/3 ) of a jot

approximately 3/8 (~= 0.374640179 ) of a temperament-unit

The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or -) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

For practical use in tuning MIDI-files, an interval's semitone value must first be calculated. The nearest integer semitone is translated into a MIDI note-number (which can generally also be described by letter-name plus optional accidental: A, Bb, C#, etc., followed by an "octave" register-number, as A-1, Bb2, etc.). Then the remainder or deficit is converted into 13mus plus or minus, respectively.

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